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<Title>Gravitational Potential</Title>
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<h1>Gravitational potential</h1>
<p>
If we compare the law of gravitation with Newton's second law, <font class="math"><b>F</b> = <i>m</i><b>a</b></font>, we see that the term
<blockquote>
<img src="gravityacc.gif">
</blockquote>
describes the <i>acceleration</i> that body 2 experiences in the gravitational field of body 1. Let's assume that body 2 is a small test mass, and body 1 is a much more massive object (consider for example the case of an artificial satellite in the gravitational field of a planet). Then we can neglect the much smaller acceleration of body 1 (the central body) in the presence of the graviational field due to body 2 (the test body).
<p>
Because the velocity of the smaller mass relative to the central body increases as a result of its acceleration in the gravitational field, the field is performing <i>work</i> on the test object, by converting <i>potential energy</i> into <i>kinetic energy</i>.
<p>
We can think of potential energy being stored in the gravitational field, and a body moving through the field from a point A to point B will change its potential energy according to the potential difference <font class="math"><i>U</i>(<b>r</b><sub><i>B</i></sub>) - <i>U</i>(<b>r</b><sub><i>A</i></sub>)</font> between the two points.
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The gravitational potential <font class="math"><i>U</i>(<b>r</b>)</font> caused by a point mass <font class="math"><i>m</i><sub>1</sub></font> at position <font class="math"><b>r</b><sub>1</sub></font> is given by
<blockquote>
<img src="gravitypot.gif">
</blockquote>
If the gravitational field is composed from multiple point masses, then the potential is simply the superposition of all contributions, denoted by the sum
<blockquote>
<img src="gravitypot2.gif">
</blockquote>
If we know the gravitational potential <font class="math"><i>U</i>(<b>r</b>)</font> at a point <font class="math"><b>r</b></font> in space, then the force of a test body of mass <font class="math"><i>m</i></font> placed at <font class="math"><b>r</b></font> can be calculated from the potential gradient:
<blockquote>
<img src="gravitypot3.gif">
</blockquote>

<img src="pot.gif"><p>
<i>Gravitational potential generated by two point sources <font class="math"><i>m</i><sub>1</sub> = 5</font> at <font class="math"><b>r</b><sub>1</sub> = (80,80)</font>, and <font class="math"><i>m</i><sub>2</sub> = 1</font> at <font class="math"><b>r</b><sub>2</sub> = (150,150)</font>. Note that the equipotential contours have been truncated around the point singularities.</i>
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<table cols=2 width=100%>
<tr><td width=50% align=center>&lt;&lt;&nbsp;<a href="Gravity.htm">Newton's law of gravitation</a></td>
<td width=50% align=center></td></tr>
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